Download On the expression of a number in the form ax2 + by + cz + du

On the expression of a number in the form
ax2 + by 2 + cz 2 + du2
Proceedings of the Cambridge Philosophical Society, XIX, 1917, 11 – 21
1. It is well known that all positive integers can be expressed as the sum of four squares.
This naturally suggests the question: For what positive integral values of a, b, c, d, can all
positive integers be expressed in the form
ax2 + by 2 + cz 2 + du2 ?
(1.1)
I prove in this paper that there are only 55 sets of values of a, b, c, d for which this is true.
The more general problem of finding all sets of values of a, b, c, d for which all integers with
a finite number of exceptions can be expressed in the form (1.1), is much more difficult and
interesting. I have considered only very special cases of this problem, with two variables
instead of four; namely, the cases in which (1.1) has one of the special forms
a(x2 + y 2 + z 2 ) + bu2 ,
(1.2)
a(x2 + y 2 ) + b(z 2 + u2 ).
(1.3)
and
These two cases are comparatively easy to discuss. In this paper I give the discussion of
(1.2) only, reserving that of (1.3) for another paper.
2. Let us begin with the first problem. We can suppose, without loss of generality, that
a ≤ b ≤ c ≤ d.
(2.1)
if a > 1, then 1 cannot be expressed in the form (1.1); and so
a=1
(2.2)
1 ≤ b ≤ 2.
(2.3)
If b > 2, then 2 is an exception; and so
We have therefore only to consider the two cases in which (1.1) has one or other of the
forms
x2 + y 2 + cz 2 + du2 , x2 + 2y 2 + cz 2 + du2 .
In the first case, If c > 3, then 3 is an exception; and so
1 ≤ c ≤ 3.
(2.31)
Paper 20
218
In the second case, if c > 5, then 5 is an exception; and so
2 ≤ c ≤ 5.
(2.32)
We can now distinguish 7 possible cases.
(2.41)
x2 + y 2 + z 2 + du2 .
If d > 7, 7 is an exception; and so
1 ≤ d ≤ 7.
(2.42)
(2.411)
x2 + y 2 + 2z 2 + du2 .
If d > 14, 14 is an exception; and so
2 ≤ d ≤ 14.
(2.43)
(2.421)
x2 + y 2 + 3z 2 + du2 .
If d > 6, 6 is an exception; and so
3 ≤ d ≤ 6.
(2.44)
(2.431)
x2 + 2y 2 + 2z 2 + du2 .
If d > 7, 7 is an exception; and so
2 ≤ d ≤ 7.
(2.45)
(2.441)
x2 + 2y 2 + 3z 2 + du2 .
If d > 10, 10 is an exception; and so
3 ≤ d ≤ 10.
(2.46)
(2.451)
x2 + 2y 2 + 4z 2 + du2 .
If d > 14, 14 is an exception; and so
4 ≤ d ≤ 14.
(2.47)
(2.461)
x2 + 2y 2 + 5z 2 + du2 .
If d > 10, 10 is an exception; and so
5 ≤ d ≤ 10.
(2.471)
We have thus eliminated all possible sets of values of a, b, c, d, except the following 55∗ :
L. E. Dickson (Bulletin of the Ameraican Math. Soc. [vol XXXIII (1927), pp. 63 – 70]) has pointed
out that Ramanujan has overlooked the fact that (1, 2, 5, 5) does not represent 15. Consequently, there are
only 54 forms.
∗
On the expression of a number in the form ax2 + by 2 + cz 2 + du2
1, 1, 1, 1
1, 2, 3, 5
1, 2, 4, 8
1, 1, 1, 2
1, 2, 4, 5
1, 2, 5, 8
1, 1, 2, 2
1, 2, 5, 5
1, 1, 2, 9
1, 2, 2, 2
1, 1, 1, 6
1, 2, 3, 9
1, 1, 1, 3
1, 1, 2, 6
1, 2, 4, 9
1, 1, 2, 3
1, 2, 2, 6
1, 2, 5, 9
1, 2, 2, 3
1, 1, 3, 6
1, 1, 2, 10
1, 1, 3, 3
1, 2, 3, 6
1, 2, 3, 10
1, 2, 3, 3
1, 2, 4, 6
1, 2, 4, 10
1, 1, 1, 4
1, 2, 5, 6
1, 2, 5, 10
1, 1, 2, 4
1, 1, 1, 7
1, 1, 2, 11
1, 2, 2, 4
1, 1, 2, 7
1, 2, 4, 11
1, 1, 3, 4
1, 2, 2, 7
1, 1, 2, 12
1, 2, 3, 4
1, 2, 3, 7
1, 2, 4, 12
1, 2, 4, 4
1, 2, 4, 7
1, 1, 2, 13
1, 1, 1, 5
1, 2, 5, 7
1, 2, 4, 13
1, 1, 2, 5
1, 1, 2, 8
1, 1, 2, 14
1, 2, 2, 5
1, 2, 3, 8
1, 2, 4, 14
1, 1, 1, 2
1, 1, 2, 4
1, 2, 4, 8
1, 1, 2, 2
1, 2, 2, 4
1, 1, 3, 3
1, 2, 2, 2
1, 2, 4, 4
1, 2, 3, 6
1, 1, 1, 4
1, 1, 2, 8
1, 2, 5, 10
1, 1, 3, 5
Of these 55 forms, the 12 forms
have been already considered by Liouville and Pepin∗ .
3. I shall now prove that all integers can be expressed in each of the 55 forms. In order to
prove this we shall consider the seven cases (2.41)–(2.47) of the previous section separately.
We shall require the following results concerning ternary quadratic arithmetical forms.
∗
There are a large number of short notes by Liouville in Vols. V–VIII of the second series of his Journal.
See also Pepin, ibid., Ser.4, Vol. VI, pp.1 – 67. The object of the work of Liouville and Pepin is rather
different from mine, viz., to determine, in a number of special cases, explicit formulæ for the number of
representations, in terms of other arithmetical functions.
219
Paper 20
220
The necessary and sufficient condition that a number cannot be expressed in the form
x2 + y 2 + z 2
(3.1)
4λ (8µ + 7), (λ = 0, 1, 2, · · · , µ = 0, 1, 2, · · ·).
(3.11)
is that it should be of the form
Similarly the necessary and sufficient conditions that a number cannot be expressed in the
forms
x2 + y 2 + 2z 2 ,
(3.2)
x2 + y 2 + 3z 2 ,
(3.3)
x2 + 2y 2 + 2z 2 ,
(3.4)
x2 + 2y 2 + 3z 2 ,
(3.5)
x2 + 2y 2 + 4z 2 ,
(3.6)
x2 + 2y 2 + 5z 2 ,
(3.7)
4λ (16µ + 14),
(3.21)
9λ (9µ + 6),
(3.31)
4λ (8µ + 7),
(3.41)
4λ (16µ + 10),
(3.51)
4λ (16µ + 14),
(3.61)
25λ (25µ + 10) or 25λ (25µ + 15).∗
(3.71)
are that it should be of the forms
Results (3.11)–(3.71) may tempt us to suppose that there are similar simple results for the form ax2 +
by + cz 2 , whatever are the values of a, b, c. It appears, however, that in most cases there are no such simple
results. For instance, the numbers which are not of the form x2 + 2y 2 + 10z 2 are those belonging to one or
other of the four classes
∗
2
25λ (8µ + 7), 25λ (25µ + 5), 25λ (25µ + 15), 25λ (25µ + 20).
Here some of the numbers of the first class belong also to one of the next three classes.
Again, the even numbers which are not of the form x2 + y 2 + 10z 2 are the numbers
4λ (16µ + 6),
while the odd numbers that are not of the form, viz.
3, 7, 21, 31, 33, 43, 67, 79, 87, 133, 217, 219, 223, 253, 307, 391, · · ·
do not seem to obey any simple law.
On the expression of a number in the form ax2 + by 2 + cz 2 + du2
221
The result concerning x2 + y 2 + z 2 is due to Cauchy: for a proof see. Landau, Handbuch
der Lehre von der Verteilung der Primzahlen, p.550. The other results can be proved in an
analogous manner. The form x2 + y 2 + 2z 2 has been considered by Lebesgue, and the form
x2 + y 2 + 3z 2 by Dirichlet. For references see Bachmann, Zahlentheorie, Vol,IV, p.149.
4. We proceed to consider the seven cases (2.41)–(2.47). In the first case we have to shew
that any number N can be expressed in the form
N = x2 + y 2 + z 2 + du2 ,
(4.1)
d being any integer between 1 and 7 inclusive.
If N is not of the form 4λ (8µ + 7), we can satisfy (4.1) with u = 0. We may therefore
suppose that N = 4λ (8µ + 7).
First, suppose that d has one of the values 1,2,4,5,6. Take u = 2λ . Then the number
N − du2 = 4λ (8µ + 7 − d)
is plainly not of the form 4λ (8µ + 7), and is therefore expressible in the form x2 + y 2 + z 2 .
Next, let d = 3. If µ = 0, take u = 2λ . Then
N − du2 = 4λ+1 .
If µ ≥ 1, take u = 2λ+1 . Then
N − du2 = 4λ (8µ − 5).
I have succeeded in finding a law in the following six simple cases:
x2 + y 2 + 4z 2 ,
x2 + y 2 + 5z 2 ,
x2 + y 2 + 6z 2 ,
x2 + y 2 + 8z 2 ,
x2 + 2y 2 + 6z 2 ,
x2 + 2y 2 + 8z 2 .
The numbers which are not of these forms are the numbers
4λ (8µ + 7) or 8µ + 3,
4λ (8µ + 3),
9λ (9µ + 3),
4λ (16µ + 14), 16µ + 6, or 4µ + 3,
4λ (8µ + 5),
4λ (8µ + 7) or 8µ + 5.
Paper 20
222
In neither of these cases is N − du2 of the form 4λ (8µ + 7), and therefore in either case it
can be expressed in the form x2 + y 2 + z 2 .
Finally, let d = 7. If µ is equal to 0,1, or 2, take u = 2λ . Then N − du2 is equal to 0, 2 · 4λ+1 ,
or 4λ+2 . If µ ≥ 3, take u = 2λ+1 . Then
N − du2 = 4λ (8µ − 21).
Therefore in either case N − du2 can be expressed in the form x2 + y 2 + z 2 .
Thus in all cases N is expressible in the form (4.1). Similarly we can dispose of the
remaining cases, with the help of the results stated in § 3. Thus in discussing (2.42) we use
the theorem that every number not of the form (3.21) can be expressed in the form (3.2).
The proofs differ only in detail, and it is not worth while to state them at length.
5. We have seen that all integers without any exception can be expressed in the form
m(x2 + y 2 + z 2 ) + nu2 ,
(5.1)
when
m = 1, 1 ≤ n ≤ 7,
and
m = 2, n = 1.
We shall now consider the values of m and n for which all integers with a finite number of
exceptions can be expressed in the form (5.1).
In the first place m must be 1 or 2. For, if m > 2, we can choose an integer ν so that
nu2 6≡ ν(mod m)
for all values of u. Then
(mµ + ν) − nu2
,
m
where µ is any positive integer, is not an integer; and so mµ + ν can certainly not be
expressed in the form (5.1).
We have therefore only to consider the two cases in which m is 1 or 2. First let us consider
the form
x2 + y 2 + z 2 + nu2 .
(5.2)
I shall shew that, when n has any of the values
1, 4, 9, 17, 25, 36, 68, 100,
(5.21)
4k + 2, 4k + 3, 8k + 5, 16k + 12, 32k + 20,
(5.22)
or is of any of the forms
On the expression of a number in the form ax2 + by 2 + cz 2 + du2
then all integers save a finite number, and in fact all integers from 4n onwards at any rate,
can be expressed in the form (5.2); but that for the remaining values of n there is an infinity
of integers which cannot be expressed in the form required.
In proving the first result we need obviously only consider numbers of the form 4λ (8µ + 7)
greater than n, since otherwise we may take u = 0. The numbers of this form less than n
are plainly among the exceptions.
6. I shall consider the various cases which may arise in order of simplicity.
(6.1)
n ≡ 0(mod 8).
There are an infinity of exceptions. For suppose that
N = 8µ + 7.
Then the number
N − nu2 ≡ 7(mod 8)
cannot be expressed in the form x2 + y 2 + z 2 .
(6.2)
n ≡ 2(mod 4).
There is only a finite number of exceptions. In proving this we may suppose that N =
4λ (8µ + 7). Take u = 1. Then the number
N − nu2 = 4λ (8µ + 7) − n
is congruent to 1, 2, 5 or 6 to modulus 8, and so can be expressed in the form x2 + y 2 + z 2 .
Hence the only numbers which cannot be expressed in the form (5.2) in this case are the
numbers of the form 4λ (8µ + 7) not exceeding n.
(6.3)
n ≡ 5(mod 8).
There is only a finite number of exceptions. We may suppose again that N = 4λ (8µ + 7).
First, let λ 6= 1. Take u = 1. Then
N − nu2 = 4λ (8µ + 7) − n ≡ 2 or 3(mod 8).
If λ = 1 we cannot take u = 1, since
N − n ≡ 7(mod 8);
so we take u = 2. Then
N − nu2 = 4λ (8µ + 7) − 4n ≡ 8(mod 32).
In either of these cases N − nu2 is of the form x2 + y 2 + z 2 .
223
Paper 20
224
Hence the only numbers which cannot be expressed in the form (5.2) are those of the form
4λ (8µ + 7) not exceeding n, and those of the form 4(8µ + 7) lying between n and 4n.
(6.4)
n ≡ 3(mod 4).
There is only a finite number of exceptions. Take
N = 4λ (8µ + 7).
if λ ≥ 1, take u = 1. Then
N − nu2 ≡ 1 or 5(mod 8).
if λ = 0, take u = 2. Then
N − nu2 ≡ 3(mod 8).
In either case the proof is completed as before.
In order to determine precisely which are the exceptional numbers, we must consider more
particularly the numbers between n and 4n for which λ = 0. For these u must be 1, and
N − nu2 ≡ 0(mod 4).
But the numbers which are multiples of 4 and which cannot be expressed in the form
x2 + y 2 + z 2 are the numbers
4κ (8ν + 7), (κ = 1, 2, 3, . . . , ν = 0, 1, 2, 3, . . .).
The exceptions required are therefore those of the numbers
n + 4κ (8ν + 7)
(6.41)
which lie between n and 4n and are of the form
8µ + 7
(6.42).
Now in order that (6.41) may be of the form (6.42), κ must be 1 if n is of the form 8k + 3,
and κ may have any of the values 2, 3, 4, . . . if n is of the form 8k + 7. Thus the only
numbers which cannot be expressed in the form (5.2), in this case, are those of the form
4λ (8µ + 7) less than n and those of the form
n + 4κ (8ν + 7),
(ν = 0, 1, 2, 3, · · ·),
lying between n and 4n, where κ = 1 if n is of the form 8k + 3, and κ > 1 if n is of the
form 8k + 7.
(6.5)
n ≡ 1(mod 8).
In this case we have to prove that
(i) if n ≥ 33, there is an infinity of integers which cannot be expressed in the form (5.2);
On the expression of a number in the form ax2 + by 2 + cz 2 + du2
(ii) if n is 1, 9, 17, or 25, there is only a finite number of exceptions.
In order to prove (i) suppose that N = 7.4λ . Then obviously u cannot be zero. But if u is
not zero u2 is always of the form 4κ (8ν + 1). Hence
N − nu2 = 7 · 4λ − n · 4κ (8ν + 1).
Since n ≥ 33, λ must be greater than or equal to κ + 2, to ensure that the right-hand side
shall not be negative. Hence
N − nu2 = 4κ (8k + 7),
where
1
k = 14 · 4λ−κ−2 − nν − (n + 7)
8
2
2
is an integer; and so N − nu is not of the form x + y 2 + z 2 .
In order to prove (ii) we may suppose, as usual, that
N = 4λ (8µ + 7).
If λ = 0, take u = 1. Then
N − nu2 = 8µ + 7 − n ≡ 6 (mod 8).
If λ ≥ 1, take u = 2λ−1 . Then
N − nu2 = 4λ−1 (8k + 3),
where
1
k = 4(µ + 1) − (n + 7).
8
In either case the proof may be completed as before. Thus the only numbers which cannot
be expressed in the form (5.2), in this case, are those of the form 8µ + 7 not exceeding n.
In other words, there is no exception when n = 1; 7 is the only exception when n = 9; 7
and 15 are the only exceptions when n = 17; 7, 15 and 23 are the only exceptions when
n = 25.
(6.6)
n ≡ 4(mod 32).
By arguments similar to those used in (6.5), we can shew that
(i) if n ≥ 132, there is an infinity of integers which cannot be expressed in the form (5.2);
(ii) if n is equal to 4, 36, 68, or 100, there is only a finite number of exceptions, namely the
numbers of the form 4λ (8µ + 7) not exceeding n.
(6.7)
n ≡ 20(mod 32).
By arguments similar to those used in (6.3), we can shew that the only numbers which
cannot be expressed in the form (5.2) are those of the form 4λ (8µ + 7) not exceeding n, and
those of the form 42 (8µ + 7) lying between n and 4n.
(6.8)
n ≡ 12(mod 16).
225
Paper 20
226
By arguments similar to those used in (6.4), we can shew that the only numbers which
cannot be expressed in the form (5.2) are those of the form 4λ (8µ + 7) less than n, and
those of the form
n + 4κ (8ν + 7), (ν = 0, 1, 2, 3, . . .),
lying between n and 4n where κ = 2 if n is of the form 4(8k + 3) and κ > 2 if n is of the
form 4(8k + 7).
We have thus completed the discussion of the form (5.2), and determined the exceptional
values of N precisely whenever they are finite in number.
7. We shall proceed to consider the form
2(x2 + y 2 + z 2 ) + nu2 .
(7.1)
In the first place n must be odd; otherwise the odd numbers cannot be expressed in this
form. Suppose then that n is odd. I shall shew that all integers save a finite number can
be expressed in the form (7.1); and that the numbers which cannot be so expressed are
(i) the odd numbers less than n,
(ii) the numbers of the form 4λ (16µ + 14) less that 4n,
(iii) the numbers of the form n + 4λ (16µ + 14) greater than n less than 9n,
(iv) the numbers of the form
cn + 4κ (16ν + 14),
(ν = 0, 1, 2, 3, . . .),
greater than 9n and less than 25n, where c = 1 if n ≡ 1 (mod 4), c = 9 if n ≡ 3 (mod 4),
κ = 2 if n2 ≡ 1 (mod 16), and κ > 2 if n2 ≡ 9 (mod 16).
First, let us suppose N even. Then, since n is odd and N is even, it is clear that u must
be even. Suppose then that
u = 2v, N = 2M.
We have to shew that M can be expressed in the form
x2 + y 2 + z 2 + 2nv 2 .
(7.2)
Since 2n ≡ 2 (mod 4), it follows from (6.2) that all integers except those which are less
than 2n and of the form 4λ (8µ + 7) can be expressed in the form (7.2). Hence the only even
integers which cannot be expressed in the form (7.1) are those of the form 4λ (16µ + 14) less
than 4n.
This completes the discussion of the case in which N is even. If N is odd the discussion
is more difficult. In the first place, all odd numbers less than n are plainly among the
exceptions. Secondly, since n and N are both odd, u must also be odd. We can therefore
suppose that
N = n + 2M, u2 = 1 + 8∆,
On the expression of a number in the form ax2 + by 2 + cz 2 + du2
227
where ∆ is an integer of the form 12 k(k + 1), so that ∆ may assume the values 0, 1, 3, 6,
. . . And we have to consider whether n + 2M can be expressed in the form
2(x2 + y 2 + z 2 ) + n(1 + 8∆),
or M in the form
x2 + y 2 + z 2 + 4n∆.
(7.3)
If M is not of the form 4λ (8µ + 7), we can take ∆ = 0. If it is of this form, and less than
4n, it is plainly an exception. These numbers give rise to the exceptions specified in (iii) of
section 7. We may therefore suppose that M is of the form 4λ (8µ + 7) and greater than 4n.
8. In order to complete the discussion, we must consider the three cases in which n ≡ 1
(mod 8), n ≡ 5 (mod 8), and n ≡ 3
(mod 4) separately.
(8.1) n ≡ 1 mod 8).
If λ is equal to 0,1, or 2, take ∆ = 1. Then
M − 4n∆ = 4λ (8µ + 7) − 4n
is of one of the forms
8ν + 3, 4(8ν + 3), 4(8ν + 6).
If λ ≥ 3 we cannot take ∆ = 1, since M − 4n∆ assumes the form 4(8ν + 7); so we take
∆ = 3. Then
M − 4n = ∆4λ (8µ + 7) − 12n
is of the form 4(8ν + 5). In either of these cases M − 4n∆ is of the form x2 + y 2 + z 2 . Hence
the only values of M , other than those already specified which cannot be expressed in the
form (7.3), are those of the form
4κ (8ν + 7),
(ν = 0, 1, 2, . . . , κ > 2),
lying between 4n and 12n. In other words, the only numbers greater than 9n which cannot
be expressed in the form (7.1), in this case, are the numbers of the form
n + 4κ (8ν + 7),
(ν = 0, 1, 2, . . . , κ > 2),
lying between 9n and 25n.
(8.2) n ≡ 5 mod 8).
if λ 6= 2, take ∆ = 1. Then
M − 4n∆ = 4λ (8µ + 7) − 4n
is of one of the forms
8ν + 3, 4(8ν + 2), 4(8ν + 3).
Paper 20
228
If λ = 2, we cannot take ∆ = 1, since M − 4n∆ assumes the form 4(8ν + 7); so we take
∆ = 3. Then
M − 4n∆ = 4λ (8µ + 7) − 12n
is of the form 4(8ν + 5). In either of these cases M − 4n∆ is of the form x2 + y 2 + z 2 . Hence
the only values of M , other than those already specified, which cannot be expressed in the
form (7.3), are those of the form 16(8µ + 7) lying between 4n and 12n. In other words, the
only numbers greater than 9n which cannot be expressed in the form (7.1), in this case, are
the numbers of the form n + 42 (16µ + 14) lying between 9n and 25n.
(8.3) n ≡ 3 (mod 4).
If λ 6= 1, take ∆ = 1. Then
M − 4n∆ = 4λ (8µ + 7) − 4n
is of one of the forms
8ν + 3, 4(4ν + 1).
If λ = 1, take ∆ = 3. Then
M − 4n∆ = 4(8µ + 7) − 12n
is of the form 4(4ν + 2). In either of these cases M − 4n∆ is of the form x2 + y 2 + z 2 .
This completes the proof that there is only a finite number of exceptions. In order to
determine what they are in this case, we have to consider the values of M , between 4n and
12n, for which ∆ = 1 and
M − 4n∆ = 4(8µ + 7 − n) ≡ 0 mod 16).
But the numbers which are multiples of 16 and which cannot be expressed in the form
x2 + y 2 + z 2 are the numbers
4κ (8ν + 7), (κ = 2, 3, 4, . . . , ν = 0, 1, 2, . . .).
The exceptional values of M required are therefore those of the numbers
4n + 4κ (8ν + 7)
(8.31)
which lie between 4n and 12n and are of the form
4(8µ + 7).
(8.32)
But in order that (8.31) may be of the form (8.32), κ must be 2 if n is of the form 8k + 3,
and κ may have any of the values 3, 4, 5, . . . if n is of the form 8k + 7. It follows that the
only numbers greater than 9n which cannot be expressed in the form (7.1), in this case, are
the numbers of the form
9n + 4κ (16ν + 14),
(ν = 0, 1, 2, . . .),
lying between 9n and 25n, where κ = 2 if n is of the form 8k + 3, and κ > 2 if n is of the
form 8k + 7.
This completes the proof of the results stated in section 7.